DISTRIBUCIÓN DE MICRO Y PEQUEÑAS EMPRESAS DEL SECTOR TEXTIL QUE TIENEN POTENCIAL EXPORTADOR SEGÚN EL NÚMERO

AC measurements on URhGe#1 at fields large enough to destroy superconductivity, but smaller than the field at which it re-enters yield intriguing results. At high (>600mK) temperatures the resistivity fits a power law as described in section4.7. At low temperatures, the resistivity drops substantially below the power law fit. The left panel of figure4.26shows several such measure- ments, similar features have also been observed on other samples[88]. It has been suggested that this might be evidence of some superconductivity surviving outside the bulk superconducting state∗

It is possible to make differential resistance measurements with AC, and the result of such measurements are shown in the right panel of figure 4.26. These measurements were made by passing a direct current through the sample (plotted on the horizontal axis) and a small AC ripple superimposed on top. When this is demodulated in the usual way the DC component disappears and the AC component measures the differential resistance of the sample over the current range of the ripple. For an ohmic sample, this should be completely constant. For a non-ohmic sample, it can change. Thus the data in figure4.26is evidence of non-linear IV curves below 500mK, and increasing non-linearities at lower temperatures. Note that as this is the derivative any deviation from constant is evidence of non-linear IV curve. So non-linearity is seen in all curves below

∗_{The low field state is known to be a bulk state from specific heat measurements[25], the high field pocket is} believed to be bulk too, but this is not proven.

0 200 400 600 800 1,000 140

160 180 200

Sample Current (µA)

Differen tal Resistance ( µ Ω) 100mK 750mK a + bI2 fit 0 200 400 600 800 1,000 −50 0 50 100 150 200

Sample Current (µA)

Sample V oltage (nV) 100mK 750mK

Figure 4.27: Directly measured IV curves of URhGe #1 in the area between the superconducting pockets. Magnetic field is 4.5 T. Left: Raw IV curves (solid lines) and the deviation from linear magnified_×10 (dotted lines). Right: Derivative of the IV curves in the left panel. Solid black lines are fits, the dashed black line is an extrapolation of the fit.

500 mK. The additional feature at about 50µA is also interesting, but that it occurs exactly when the AC ripple is equal to the direct current is a suspicious coincidence, and there is no equivalent feature in the DC measurements.

This AC technique is not ideal however. Whilst in theory one could regain the IV curve by integrating, there are several possible pitfalls. Firstly, the differential resistance is inevitably averaged over the size of the ripple voltage — in this case 50µArms. Secondly, it is difficult to

be sure the result is not due to some form of relaxation in the sample on time constants similar to the measurement frequency, as might arise from a flux lattice. The SQUID bridge also offers excellent signal to noise characteristics.

In order to further investigate this effect, IV curves were measured using the SQUID resistance bridge detailed in section 3.4. In all cases, the deviation from linearity is very small, never exceeding 5 nV. It is nonetheless present; IV curves are shown in figure4.27.

The IV curves were measured by sweeping the current continuously from 0 to 1 mA at 1µA/s. Each curve was measured for positive and negative currents, to eliminate thermoelectric contri- butions due to sample heating. Such contributions are two orders of magnitude smaller than the observed nonlinearity, never exceeding 50 pV. Finally, a small correction is made to reflect that the bridge has a response time of about 1 s (due to the PID controller settings), so with a continuous sweeping sample current the observed voltage, calculated from the reference current, is always slightly behind. For a ramp rate of 1µA/s, an offset of 1µA would be expected, and an adjustment of 1.05µA was found to make the IV curve smooth and continuous at the origin. This again is much smaller than the scale of the features observed.

To better understand the features, the differential resistance d_dV_I is plotted in the right panel of figure 4.27. This is calculated by fitting a straight line over a sliding 30µA window. The 750 mK data, assumed to be at a temperature well above any superconducting effects, is not the flat line one would expect for a normal metal. This can be understood as the result of heating. One might reasonably assume that for small temperature changes, R = R0+ d_dR_T

_T

0(T −T0) and, as the temperature change is due to heating, T =T0+K·I2Rcts where Rcts is the series

resistance of the two current contacts. This leads to a resistance ofR=R0+ ddRT _T

0K·I 2_Rcts.

Finally we can also write d_dV_I = _dd_IIR =R0+ 1₃d_dR_TT 0K·I

2_{Rcts. This then is the motivation}

for the black line fit in figure4.27. We can estimateRcts as 100 mΩ from direct measurements of a small number of similar spot welds at 2 K, and d_dR_T

_{750 mK} as 80 nΩ/mK from the left panel of figure4.26∗. This allows us to calculate the strength of the thermal link as 1.4µW/K, a value which is eight times that which one would expect from the Wiedmann-Franz law if heat flow is via the contacts, but which seems reasonable given that the contacts do not lead directly to the bath. The non-linearity of the 750 mK IV curve can then be entirely explained by heating. It should be noted that at 1 mA the sample would be heated by 74 mK, which is 10% of the absolute temperature, but as the temperature dependent resistivity is fairly smooth, our linearisation of R(T) is still valid.

A similar analysis is possible on the 100 mK curve, fitting only to the data above 290µA. The gradient d_dR_T

_{100 mK} is larger at 142 nΩ/mK, which yields K = 2.5µW/K. This is a little larger than above, which is surprising given the lower temperature. This suggests that heating is not the cause, and it is in fact a property of the normal state resistivity. One possible cause of this arises from the small and irregularly shaped sample, and the presence of field. If the carriers entering the sample spread out from the contacts, their paths are constrained by their orbital motion in the field. This effect is known as current jetting, and can be very complicated and unpredictable in non bar-shaped samples. It is also not proven that the SQUID bridge is perfectly linear in current. The deviation is well outside the specified linearity of the current supplies, and non-linearity of the SQUID should not enter as it is used as a null detector. Ultimately however, whatever the source of the non-linearity it is very similar in both curves so can be identified as some feature of the normal state resistance.

The sharp downturn below 200µA is however a clear and distinctive feature. If we discount the curvature mentioned above, then what remains is a differential resistance curve which rises approximately linearly (with non-zero intercept) at low currents, then is flat above 250µA. The corresponding IV curve is linear at high I, but curves upwards slightly at low I. A reduced voltage at low current is possibly an indicator that there is some form or superconductivity and a critical current. This is broadly in keeping with the expected form discussed in section1.4.2if the critical current so small that the rounding off of the transition extends all the way to zero,

∗_{A rescaling of 10% is necessary to make the AC measurement match the gradient of the IV curve. This is} due to the uncalibrated reference resistor

and if the superconducting material does not form a complete path through the sample.

When considering the possible form of any superconductivity in this region, it is prudent to start with the nature of the low-field pocket. In the limit of weak pinning, one can see a transition from a zero resistance state to one with resistance atHc1, as flux enters the sample. This cannot

be the case in URhGe, as its ferromagnetic nature ensures thatB is always large enough to have flux in the sample. This is confirmed by torque magnetization measurements on superconducting single crystals[88] which show no evidence for a Meissner state. That leaves two options for the transition, either it isHc2, and the bulk superconducting state is destroyed, or it is a transition

from a pinned flux lattice to flux flow.

The former can explain why the deviation from linearity is small. Any remaining supercon- ductivity would be confined to surfaces, defects or domain wall boundariesetc. If these filaments do not join up, the majority of the current path is normal. Many small filaments is also a possible explanation for the lack of clear single critical current. If the filaments are randomly distributed through the material, they will experience different current densities and thus transition at dif- ferent times. If they are associated with different forms of lattice defect (for example) they may also have different critical currents. This then provides a method by which the observations could be due to some form of superconductivity, but it is far from conclusive evidence of such.

Another option that should be considered is superconducting fluctuations near the supercon- ducting state. If small droplets of superfluid can appear and disappear, without being able to nucleate a complete transition to superconductivity, then they can reduce the overall resistance. Tinkham[13] gives a rough estimate of the increase in conductivity from such fluctuations as:

σ= 1 32 e2 ~ξ(0) _T T₋Tc 1/2 (4.9)

Whereξ(0), the coherence length at zero temperature, is possibly very small in URhGe (see the arguments in section 4.5.2). However, if one estimates a coherence length of 10−9_{m from the}

orbital limiting field in [39], this still only contributes a conductivity that is less than one part in 104_{of the measured conductivity. So it is considered that any contribution from superconducting}

fluctuations should be negligible.

In conclusion, the small deviation from the normal resistance; the previous measurements that show the transition from the zero resistance state is of the form expected for Hc2; and

the low critical currents strongly suggest that these new observations are of a non-bulk state. Several forms of non-bulk superconductivity are possible, but domain-wall superconductivity is an excellent fit for the other known properties of the material. For field applied close to thebaxis direction, and below the moment rotation transition, the material is ferromagnetic with domains where the moment is aligned in the_±c direction. Superconductivity is believed to be destroyed by orbital limiting in URhGe, and there will be some domain walls where the field lies in the

0 50 100 150 200 0 10 20 30 40 50 Current [µA] Resistance [ µ Ω]

Figure 4.28: DC resistance of URhGe sample MK3 measured at zero field close to the super- conducting transition. Curves at 5 mK intervals from 235 mK (bottom, black circles) to 295 mK (top, cyan circles).

plane of the domain wall. These will continue to superconduct at temperatures up to the Tc arising from the effect of field as a tuning parameter. As such, it is considered highly likely that these observations are of domain wall superconductivity. TheTc at which these effects disappear could be useful in identifying what the bulkTc would be if B were simply a tuning parameter rather than also acting through orbital limiting.